In our further considerations, we need the following lemmas.
Lemma 2.1
(Shao [13])
Let
\(\{X_{i}, 1\leq i \leq n\}\)
be a sequence of negatively associated random variables with
\(EX_{i}=0\)
and
\(E|X_{i}|^{q}<\infty\)
for
\(q>1\)
and every
\(1\leq i\leq n\). Then
$$ E\max_{1\leq k\leq n}\Biggl|\sum_{i=1}^{k} X_{i}\Biggr|^{q}\leq2^{3-q}\sum _{i=1}^{n} E|X_{i}|^{q} \quad \textit{for } 1< q\leq2 $$
(2.1)
and
$$ E\max_{1\leq k\leq n}\Biggl|\sum_{i=1}^{k} X_{i}\Biggr|^{q}\leq2\biggl(\frac{15 q}{\ln q}\biggr)^{q} \Biggl\{ \sum_{i=1}^{n} E|X_{i}|^{q}+ \Biggl(\sum_{i=1}^{n}EX_{i}^{2} \Biggr)^{\frac{q}{2}}\Biggr\} \quad\textit{for } q> 2. $$
(2.2)
Remark
In Lemma 2.1 for \(q=1\), we have
$$E\max_{1\leq k\leq n}\Biggl|\sum_{i=1}^{k} X_{i}\Biggr|\leq\sum_{i=1}^{n} E|X_{i}|. $$
Lemma 2.2
(Gut [7])
Let
\(\{X_{n}, n\geq1\}\)
be a sequence of random variables satisfying a weak mean dominating condition with mean dominating random variable
X, that is, for some
\(c>0\), (1.2) holds for all
\(x>0\)
and
\(n\geq1\).
Let
\(r>0\)
and, for some
\(A>0\),
$$\begin{aligned}& X_{i}^{\prime}=X_{i} I\bigl(|X_{i}|\leq A\bigr), \qquad X_{i}^{\prime\prime}=X_{i} I\bigl(|X_{i}|>A\bigr), \\& X_{i}^{*}=X_{i} I\bigl(|X_{i}|\leq A\bigr)-AI(X_{i}< -A)+AI(X_{i}>A) \end{aligned}$$
and
$$\begin{aligned}& X^{\prime}=X I\bigl(|X|\leq A\bigr), \qquad X^{\prime\prime}=X I\bigl(|X|>A\bigr), \\& X^{*}=X I\bigl(|X|\leq A\bigr)-AI(X< -A)+AI(X>A) . \end{aligned}$$
Then, for some constant
\(C>0\),
-
(i)
if
\(E|X|^{r}<\infty\), then
\((n^{-1})\sum_{i=1}^{n} E|X_{i}|^{r}\leq CE|X|^{r}\),
-
(ii)
\((n^{-1})\sum_{i=1}^{n} E|X_{i}^{\prime}|^{r}\leq C(E|X^{\prime}|^{r}+A^{r} P(|X|>A))\)
for all
\(A>0\),
-
(iii)
\((n^{-1})\sum_{i=1}^{n} E|X_{i}^{\prime\prime}|^{r}\leq CE|X^{\prime\prime}|^{r}\)
for all
\(A>0\),
-
(iv)
\((n^{-1})\sum_{i=1}^{n} E|X_{i}^{*}|^{r}\leq CE|X^{*}|^{r}\)
for all
\(A>0\).
The following result is obtained by Theorem 2.1 of Kuczmaszewska [1].
Lemma 2.3
(Kuczmaszewska [1])
Let
\(\alpha p>1\), \(p>0\), and
\(\alpha>\frac{1}{2}\). Let
\(\{X_{n}, n\geq1\}\)
be a sequence of negatively associated random variables with
\(EX_{n}=0\)
for all
\(n\geq 1\), and
X
be a random variable possibly defined on a different space satisfying condition (1.2) for all
\(\epsilon>0\)
and
\(n\geq1\). Then
$$ E|X|^{p}< \infty $$
(2.3)
implies
$$ \sum_{n=1}^{\infty}n^{\alpha p-2}P\Biggl( \max_{1\leq j\leq n}\Biggl|\sum_{i=1}^{j} X_{i}\Biggr|>\epsilon n^{\alpha}\Biggr)< \infty \quad\textit{for all } \epsilon>0. $$
(2.4)
Lemma 2.4
Let
\(\alpha p>1\), \(\alpha>\frac{1}{2}\), and
\(p>0\). Let
\(\{X_{n}, n\geq1\}\)
be a sequence of negatively associated random variables with
\(EX_{n}=0\)
for all
\(n\geq1\), and
X
be a random variable possibly defined on a different space satisfying condition (1.2) for all
\(\epsilon>0\)
and
\(n\geq1\). Then (2.3) implies
$$ \sum_{n=1}^{\infty}n^{\alpha p-2-\alpha} \int_{n^{\alpha}}^{\infty}P\Biggl(\max_{1\leq k\leq n}\Biggl| \sum_{i=1}^{k} X_{i}\Biggr|>u\Biggr)\,du< \infty. $$
(2.5)
Proof
In the case \(0< p<1\), let, for all \(u>0\),
$$X_{i}=X_{i}I\bigl[|X_{i}|\leq u\bigr]+X_{i}I\bigl[|X_{i}|> u\bigr]=X_{ui}+X_{ui}^{\prime} $$
and
$$S_{n}=\sum_{i=1}^{n} X_{i}I\bigl[|X_{i}|\leq u\bigr]+\sum_{i=1}^{n} X_{i}I\bigl[|X_{i}|>u\bigr]=\sum_{i=1}^{n} X_{ui}+\sum_{i=1}^{n} X_{ui}^{\prime}. $$
Then we obtain
$$\begin{aligned} &\sum_{n=1}^{\infty}n^{\alpha p-2-\alpha} \int_{n^{\alpha}}^{\infty}P\Bigl(\max_{1\leq j\leq n}|S_{j}|>u \Bigr)\,du \\ &\quad\leq\sum_{n=1}^{\infty}n^{\alpha p-2-\alpha} \int_{n^{\alpha}}^{\infty}P\Biggl(\max_{1\leq j\leq n}\Biggl| \sum_{i=1}^{j} X_{ui}\Biggr|> \frac{u}{2}\Biggr)\,du \\ &\qquad{} +\sum_{n=1}^{\infty}n^{\alpha p-2-\alpha} \int_{n^{\alpha}}^{\infty}P\Biggl(\max_{1\leq j\leq n}\Biggl| \sum_{i=1}^{j} X_{ui}^{\prime}\Biggr|> \frac{u}{2}\Biggr)\,du \\ &\quad:=I+J. \end{aligned}$$
(2.6)
For J, take \(q=p\) (\(0< p<1\)). By the Markov inequality, Lemma 2.2(iii), and \(E|X|^{p}<\infty\) we have that
$$\begin{aligned} J \leq&C\sum_{n=1}^{\infty}n^{\alpha p-2-\alpha} \int_{n^{\alpha}}^{\infty}u^{-\frac{q}{2}}E\Biggl(\max _{1\leq j\leq n}\Biggl|\sum_{i=1}^{j} X_{ui}^{\prime}\Biggr|\Biggr)^{\frac {q}{2}}\,du \quad \mbox{by Markov inequality} \\ \leq&C\sum_{n=1}^{\infty}n^{\alpha p-2-\alpha} \int_{n^{\alpha}}^{\infty}u^{-\frac{q}{2}}\sum _{i=1}^{n} E\bigl|X_{ui}^{\prime}\bigr|^{\frac{q}{2}}\,du\quad \biggl(\frac {q}{2}< \frac{1}{2}\biggr) \\ \leq&C\sum_{n=1}^{\infty}n^{\alpha p-2-\alpha} \int_{n^{\alpha}}^{\infty}u^{-\frac{q}{2}}\sum _{i=1}^{n} E|X_{i}|^{\frac{q}{2}}I\bigl(|X_{i}|>u\bigr)\,du \\ \leq&C\sum_{n=1}^{\infty}n^{\alpha p-1-\alpha} \int_{n^{\alpha}}^{\infty}u^{-\frac{q}{2}}E|X|^{\frac{q}{2}}I\bigl(|X|>u\bigr)\,du \quad \mbox{by Lemma}~2.2(\mathrm{iii}) \\ =&C\sum_{n=1}^{\infty}n^{\alpha p-1-\alpha}\sum _{m=n}^{\infty}\int _{m^{\alpha}}^{(m+1)^{\alpha}} u^{-\frac{p}{2}}E|X|^{\frac {p}{2}}I\bigl(|X|>u\bigr)\,du\quad (q=p) \\ \leq&C\sum_{n=1}^{\infty}n^{\alpha p-1-\alpha}\sum _{m=n}^{\infty}\int _{m^{\alpha}}^{(m+1)^{\alpha}} m^{-\frac{\alpha p}{2}}E|X|^{\frac {p}{2}}I\bigl(|X|>u\bigr)\,du \\ \leq&C\sum_{n=1}^{\infty}n^{\alpha p-1-\alpha}\sum _{m=n}^{\infty}\int _{m^{\alpha}}^{(m+1)^{\alpha}} m^{-\frac{\alpha p}{2}}E|X|^{\frac {p}{2}}I \bigl(|X|>m^{\alpha}\bigr)\,du \\ \leq&C\sum_{n=1}^{\infty}n^{\alpha p-1-\alpha}\sum _{m=n}^{\infty}m^{-\frac{\alpha p}{2}+\alpha-1}E|X|^{\frac{p}{2}}I \bigl(|X|>m^{\alpha}\bigr) \\ =&C\sum_{m=1}^{\infty}m^{-\frac{\alpha p}{2}+\alpha-1}E|X|^{\frac {p}{2}}I \bigl(|X|>m^{\alpha}\bigr)\sum_{n=1}^{m} n^{\alpha p-1-\alpha} \\ \leq&C\sum_{m=1}^{\infty}m^{\frac{\alpha p}{2}-1} E|X|^{\frac {p}{2}}I\bigl(|X|>m^{\alpha}\bigr) \\ =&C\sum_{m=1}^{\infty}m^{\frac{\alpha p}{2}-1}\sum _{n=m}^{\infty}E|X|^{\frac{p}{2}}I \bigl(n^{\alpha}< |X|\leq(n+1)^{\alpha}\bigr) \\ \leq&C\sum_{n=1}^{\infty}E|X|^{\frac{p}{2}}I \bigl(n^{\alpha}< |X|\leq (n+1)^{\alpha}\bigr)\sum _{m=1}^{n} m^{\frac{\alpha p}{2}-1} \\ \leq&C\sum_{n=1}^{\infty}n^{\frac{\alpha p}{2}} E|X|^{\frac {p}{2}}I\bigl(n^{\alpha}< |X|\leq(n+1)^{\alpha}\bigr) \\ \leq&C E|X|^{p}< \infty. \end{aligned}$$
(2.7)
Similarly, for I, take \(q=p\) (\(0< p<1\)). By the Markov inequality and Lemma 2.2(ii) we have that
$$\begin{aligned} I \leq&C\sum_{n=1}^{\infty}n^{\alpha p-2-\alpha} \int_{n^{\alpha}}^{\infty}u^{-\frac{q}{2}}E\Biggl(\max _{1\leq j\leq n}\Biggl|\sum_{i=1}^{j} X_{ui}\Biggr|\Biggr)^{\frac {q}{2}}\,du \quad \mbox{by Markov inequality} \\ \leq&C\sum_{n=1}^{\infty}n^{\alpha p-2-\alpha} \int_{n^{\alpha}}^{\infty}u^{-\frac{p}{2}}\sum _{i=1}^{n} E|X_{ui}|^{\frac{p}{2}}\,du \quad \biggl(q=p \mbox{ and } \frac{p}{2}< \frac{1}{2}\biggr) \\ \leq&C\sum_{n=1}^{\infty}n^{\alpha p-2-\alpha} \int_{n^{\alpha}}^{\infty}u^{-\frac{p}{2}}\sum _{i=1}^{n} \bigl\{ E|X_{i}|^{\frac{p}{2}}I\bigl(|X_{i}| \leq u\bigr)\bigr\} \,du \\ \leq&C\sum_{n=1}^{\infty}n^{\alpha p-1-\alpha} \int_{n^{\alpha}}^{\infty}u^{-\frac{p}{2}}E|X|^{\frac{p}{2}}I\bigl(|X| \leq u\bigr)\,du \\ &{} +C\sum_{n=1}^{\infty}n^{\alpha p-1-\alpha} \int_{n^{\alpha}}^{\infty}P\bigl(|X|>u\bigr)\,du \quad\mbox{by Lemma}~2.2(\mathrm{ii}) \\ \leq&C\sum_{n=1}^{\infty}n^{\alpha p-1-\alpha} \int_{n^{\alpha}}^{\infty}u^{-\frac{p}{2}}E|X|^{\frac{p}{2}}I\bigl(|X| \leq u\bigr)\,du \\ &{}+C\sum_{n=1}^{\infty}n^{\alpha p-1-\alpha} \int_{n^{\alpha}}^{\infty}u^{-\frac{p}{2}}E|X|^{\frac{p}{2}}I\bigl(|X|>u\bigr)\,du \\ =&I_{1}+I_{2}. \end{aligned}$$
In the processing of (2.7), we obtain \(I_{2}<\infty\). It remains to prove that \(I_{1}<\infty\):
$$\begin{aligned} I_{1} =&C\sum_{n=1}^{\infty}n^{\alpha p-1-\alpha}\sum_{m=n}^{\infty}\int _{m^{\alpha}}^{(m+1)^{\alpha}}u^{-\frac{p}{2}}E|X|^{\frac{p}{2}}I\bigl(|X| \leq u\bigr)\,du \\ \leq&C\sum_{n=1}^{\infty}n^{\alpha p-1-\alpha}\sum _{m=n}^{\infty}m^{-\frac{\alpha p}{2}+\alpha-1} E|X|^{\frac{p}{2}}I\bigl(|X|\leq (m+1)^{\alpha}\bigr) \\ =&C\sum_{m=1}^{\infty}m^{-\frac{\alpha p}{2}+\alpha-1} E|X|^{\frac {p}{2}}I\bigl(|X|\leq(m+1)^{\alpha}\bigr)\sum _{n=1}^{m} n^{\alpha p-1-\alpha} \\ \leq&C\sum_{m=1}^{\infty}m^{\frac{\alpha p}{2}-1}E|X|^{\frac {p}{2}}I \bigl(|X|\leq(m+1)^{\alpha}\bigr) \\ \leq&C\sum_{m=1}^{\infty}m^{\frac{\alpha p}{2}-1} \sum_{n\leq m} E|X|^{\frac{p}{2}}I\bigl(n^{\alpha}< |X| \leq(n+1)^{\alpha}\bigr) \\ \leq&C\sum_{m=1}^{\infty}m^{\frac{\alpha p}{2}-1} \sum_{n\leq m}n^{\frac {\alpha p}{2}}P\bigl(n^{\alpha}< |X| \leq(n+1)^{\alpha}\bigr) \\ \leq&C\sum_{m=1}^{\infty}m^{\alpha p}P \bigl(m^{\alpha}< |X|\leq(m+1)^{\alpha}\bigr) \leq CE|X|^{p}< \infty. \end{aligned}$$
(2.8)
Hence, from (2.6)-(2.8) the result (2.5) follows in the case \(0< p<1\).
In the case \(1\leq p < 2\), let \(Y_{ui}=X_{i}I(|X_{i}|\leq u)-uI(X_{i} < -u)+uI(X_{i}>u)\) for all \(u>0\), and \(Y_{ui}^{\prime}=X_{i}-Y_{ui}\), \(i\geq1\). Then we have
$$\begin{aligned} &\sum_{n=1}^{\infty}n^{\alpha p-2-\alpha} \int_{n^{\alpha}}^{\infty}P\Biggl(\max_{1\leq k\leq n}\Biggl| \sum_{i=1}^{k} X_{i}\Biggr|>u\Biggr)\,du \\ &\quad\leq\sum_{n=1}^{\infty}n^{\alpha p-2-\alpha} \int_{n^{\alpha}}^{\infty}P\Biggl(\max_{1\leq k\leq n}\Biggl| \sum_{i=1}^{k} (Y_{ui}-EY_{ui})\Biggr|> \frac{u}{2}\Biggr)\,du \\ &\qquad{} +\sum_{n=1}^{\infty}n^{\alpha p-2-\alpha} \int_{n^{\alpha}}^{\infty}P\Biggl(\max_{1\leq k\leq n}\Biggl| \sum_{i=1}^{k}\bigl(Y_{ui}^{\prime}-EY_{ui}^{\prime} \bigr)\Biggr|>\frac {u}{2}\Biggr)\,du \quad\mbox{since }EX_{n}=0 \\ &\quad:=I^{\prime}+J^{\prime}. \end{aligned}$$
(2.9)
Note that \(\{Y_{ui}-EY_{ui}\}\) and \(\{Y_{ui}^{\prime}-EY_{ui}^{\prime}\}\) are sequences of negatively associated random variables.
For \(J^{\prime}\), take q such that \(1\leq p< q\leq2\). By the fact that \(|Y_{ui}^{\prime}|\leq|X_{i}|I(|X_{i}|>u)\), the Markov inequality, Lemma 2.1, (2.1), Lemma 2.2(iii), (2.3), and the \(C_{r}\)-inequality we have
$$\begin{aligned} J^{\prime} \leq&C\sum_{n=1}^{\infty}n^{\alpha p-2-\alpha} \int_{n^{\alpha}}^{\infty}u^{-q} \sum _{i=1}^{n} E\bigl|Y_{ui}^{\prime}-EY_{ui}^{\prime}\bigr|^{q}\,du \quad\mbox{by}~(2.1) \\ \leq&C\sum_{n=1}^{\infty}n^{\alpha p-2-\alpha} \int_{n^{\alpha}}^{\infty}u^{-q}\sum _{i=1}^{n} E\bigl|Y_{ui}^{\prime}\bigr|^{q}\,du \quad\mbox{(by the $C_{r}$-inequality)} \\ \leq&C\sum_{n=1}^{\infty}n^{\alpha p-2-\alpha} \int_{n^{\alpha}}^{\infty}u^{-q}\sum _{i=1}^{n} E|X_{i}|^{q}I\bigl(|X_{i}|>u\bigr)\,du \\ \leq&C\sum_{n=1}^{\infty}n^{\alpha p-1-\alpha} \int_{n^{\alpha}}^{\infty}u^{-q}E|X|^{q}I\bigl(|X|>u\bigr)\,du \quad\mbox{Lemma}~2.2(\mathrm{iii}) \\ =&C\sum_{n=1}^{\infty}n^{\alpha p-1-\alpha}\sum _{m=n}^{\infty}\int _{m^{\alpha}}^{(m+1)^{\alpha}} u^{-q}E|X|^{q}I\bigl(|X|>u\bigr)\,du \\ \leq&C\sum_{n=1}^{\infty}n^{\alpha p-1-\alpha}\sum _{m=n}^{\infty}m^{-\alpha q+\alpha-1}E|X|^{q}I \bigl(|X|>m^{\alpha}\bigr) \\ \leq&C\sum_{m=1}^{\infty}m^{-\alpha q+\alpha-1}E|X|^{q}I \bigl(|X|>m^{\alpha}\bigr)\sum_{n=1}^{m} n^{\alpha p-1-\alpha} \\ =&C\sum_{m=1}^{\infty}m^{-\alpha(q-p)-1} E|X|^{q}I\bigl(|X|>m^{\alpha}\bigr) \\ =&C\sum_{m=1}^{\infty}m^{-\alpha(q-p)-1}\sum _{n=m}^{\infty}E|X|^{q}I \bigl(n^{\alpha}< |X|\leq(n+1)^{\alpha}\bigr) \\ \leq&C\sum_{n=1}^{\infty}E|X|^{q}I \bigl(n^{\alpha}< |X|\leq(n+1)^{\alpha}\bigr)\sum _{m=1}^{n} m^{-\alpha(q-p)-1} \\ \leq&C\sum_{n=1}^{\infty}E|X|^{p}I \bigl(n^{\alpha}< |X|\leq(n+1)^{\alpha}\bigr) \\ \leq&C E|X|^{p}< \infty. \end{aligned}$$
(2.10)
For \(I^{\prime}\), take q such that \(1< p< q\leq2\). By the Markov inequality, (1.2), Lemma 2.1, (2.1), the \(C_{r}\)-inequality, and Lemma 2.2(iv) we have
$$\begin{aligned} I^{\prime} \leq&C\sum_{n=1}^{\infty}n^{\alpha p-2-\alpha} \int_{n^{\alpha}}^{\infty}u^{-q}E\Biggl(\max _{1\leq k\leq n}\sum_{i=1}^{k}|Y_{ui}-EY_{ui}|^{q} \Biggr)\,du \quad\mbox{by Markov inequality} \\ \leq&C\sum_{n=1}^{\infty}n^{\alpha p-2-\alpha} \int_{n^{\alpha}}^{\infty}u^{-q}\sum _{i=1}^{n} E|Y_{ui}-EY_{ui}|^{q} \,du \quad\mbox{by}~(2.1) \\ \leq&C\sum_{n=1}^{\infty}n^{\alpha p-2-\alpha} \int_{n^{\alpha}}^{\infty}u^{-q}\sum _{i=1}^{n} E|Y_{ui}|^{q}\,du \quad \mbox{by the $C_{r}$-inequality} \\ \leq&C\sum_{n=1}^{\infty}n^{\alpha p-1-\alpha} \int_{n^{\alpha}}^{\infty}u^{-q}E|X|^{q}I\bigl(|X| \leq u\bigr)\,du \\ &{} +C\sum_{n=1}^{\infty}n^{\alpha p-1-\alpha} \int_{n^{\alpha}}^{\infty}u^{-q}E|X|^{q}I\bigl(|X|>u\bigr)\,du \quad \mbox{by Lemma}~2.2(\mathrm{iv}) \\ =&I_{1}^{\prime}+I_{2}^{\prime}. \end{aligned}$$
According to the calculation of \(J^{\prime}\), we obtain \(I_{2}^{\prime}<\infty\) (see (2.10)). It remains to prove that \(I_{1}^{\prime}<\infty\). By taking q such that \(1\leq p< q\leq2\) we have
$$\begin{aligned} I_{1}^{\prime} =&C\sum _{n=1}^{\infty}n^{\alpha p-1-\alpha}\sum _{m=n}^{\infty}\int _{m^{\alpha}}^{(m+1)^{\alpha}}u^{-q}E|X|^{q}I\bigl(|X| \leq u\bigr)\,du \\ \leq&C\sum_{n=1}^{\infty}n^{\alpha p-1-\alpha}\sum _{m=n}^{\infty}m^{-\alpha q+\alpha-1} E|X|^{q}I\bigl(|X|\leq(m+1)^{\alpha}\bigr) \\ =&C\sum_{m=1}^{\infty}m^{-\alpha q+\alpha-1} E|X|^{q}I\bigl(|X|\leq (m+1)^{\alpha}\bigr)\sum _{n=1}^{m} n^{\alpha p-1-\alpha} \\ \leq&C\sum_{m=1}^{\infty}m^{-\alpha q+\alpha p-1}E|X|^{q}I \bigl(|X|\leq (m+1)^{\alpha}\bigr) \\ \leq&C\sum_{m=1}^{\infty}m^{-\alpha q+\alpha p-1}\sum _{n\leq m} E|X|^{q}I\bigl(n^{\alpha}< |X| \leq(n+1)^{\alpha}\bigr) \\ \leq&C\sum_{m=1}^{\infty}m^{\alpha p}P \bigl(m^{\alpha}< |X|\leq(m+1)^{\alpha}\bigr) \\ =&CE|X|^{p}< \infty. \end{aligned}$$
(2.11)
Hence, from (2.9)-(2.11) the result (2.5) follows in the case \(1\leq p< 2\).
In the case \(p\geq2\), we also obtain \(I^{\prime}\) and \(J^{\prime}\) of (2.9).
For \(I^{\prime}\), by the Markov inequality, the \(C_{r}\)-inequality, and Lemma 2.1 (2.2) we have that, for \(q>2\),
$$\begin{aligned} I^{\prime} \leq&C\sum_{n=1}^{\infty}n^{\alpha p-2-\alpha} \int_{n^{\alpha}}^{\infty}u^{-q}E\Biggl\{ \max _{1\leq j\leq n}\Biggl|\sum_{i=1}^{j}(Y_{ui}-EY_{ui})\Biggr| \Biggr\} ^{q} \,du \\ \leq&C\sum_{n=1}^{\infty}n^{\alpha p-2-\alpha} \int_{n^{\alpha}}^{\infty}u^{-q}\Biggl\{ \sum _{i=1}^{n}E|Y_{ui}-EY_{ui}|^{q} +\Biggl(\sum_{i=1}^{n}E|Y_{ui}-EY_{ui}|^{2} \Biggr)^{\frac{q}{2}}\Biggr\} \,du \quad\mbox{by}~(2.2) \\ \leq&C\sum_{n=1}^{\infty}n^{\alpha p-2-\alpha} \int_{n^{\alpha}}^{\infty}u^{-q}\sum _{i=1}^{n}E|Y_{ui}|^{q} \,du \\ &{} +C\sum_{n=1}^{\infty}n^{\alpha p-2-\alpha} \int_{n^{\alpha}}^{\infty}u^{-q}\Biggl(\sum _{i=1}^{n} EY_{ui}^{2} \Biggr)^{\frac{q}{2}}\,du \quad\mbox{by the $C_{r}$-inequality} \\ =&I_{3}^{\prime}+I_{4}^{\prime}. \end{aligned}$$
(2.12)
We will consider \(I_{3}^{\prime}\) and \(I_{4}^{\prime}\) as follows.
Note that \(\alpha>\frac{1}{2}\), \(\alpha p>1\), and \(p\geq2\). Take \(q>\max(p, \frac{\alpha p-1}{\alpha-\frac{1}{2}})\), which implies that \(\alpha p-2-\alpha q+\frac{q}{2}<-1\).
By Lemma 2.2(iv) we have
$$\begin{aligned} I_{3}^{\prime} \leq&C\sum_{n=1}^{\infty}n^{\alpha p-1-\alpha} \int_{n^{\alpha}}^{\infty}u^{-q}E|X|^{q}I\bigl(|X| \leq u\bigr)\,du \\ &{}+C\sum_{n=1}^{\infty}n^{\alpha p-1-\alpha} \int_{n^{\alpha}}^{\infty}u^{-q}E|X|^{q}I\bigl(|X|> u\bigr)\,du \\ \leq&I_{31}^{\prime}+I_{32}^{\prime}. \end{aligned}$$
According to the calculation of \(J^{\prime}\), we obtain \(I_{32}^{\prime}<\infty\) (see (2.10)). It remains to prove \(I_{31}^{\prime}\):
$$\begin{aligned} I_{31}^{\prime} \leq&C\sum _{n=1}^{\infty}n^{\alpha p-1-\alpha}\sum _{m=n}^{\infty}\int_{m^{\alpha}}^{(m+1)^{\alpha}} u^{-q}E|X|^{q}I\bigl(|X| \leq u\bigr)\,du \\ \leq&C\sum_{n=1}^{\infty}n^{\alpha p-1-\alpha}\sum _{m=n}^{\infty}m^{\alpha-1-\alpha q}E|X|^{q}I \bigl(|X|\leq(m+1)^{\alpha}\bigr) \\ =&C\sum_{m=1}^{\infty}m^{\alpha-1-\alpha q}E|X|^{q}I \bigl(|X|\leq(m+1)^{\alpha}\bigr)\sum_{m=1}^{n} m^{\alpha p-1-\alpha} \\ \leq&C\sum_{n=1}^{\infty}n^{\alpha p-1-\alpha q}E|X|^{q}I \bigl(|X|\leq (m+1)^{\alpha}\bigr) \\ \leq&C\sum_{m=1}^{\infty}m^{\alpha p-1-\alpha q}E|X|^{q}I \bigl(m^{\alpha}< |X|\leq(m+1)^{\alpha}\bigr) \\ &{} +C\sum_{m=1}^{\infty}m^{\alpha p-1-\alpha q}E|X|^{q}I \bigl(|X|\leq m^{\alpha}\bigr) \\ \leq&C\sum_{m=1}^{\infty}m^{-1} E|X|^{p}I\bigl(m^{\alpha}< |X|\leq(m+1)^{\alpha}\bigr) \\ &{} +C\sum_{m=1}^{\infty}m^{-\alpha(q-p)-1}\sum _{j=1}^{m} j^{\alpha q}P \bigl((j-1)^{\alpha}< |X|\leq j^{\alpha}\bigr) \\ \leq&C\sum_{m=1}^{\infty}E|X|^{p}I \bigl(m^{\alpha}< |X|\leq(m+1)^{\alpha}\bigr) \\ &{} +C\sum_{j=1}^{\infty}j^{\alpha q}P \bigl((j-1)^{\alpha}< |X|\leq j^{\alpha}\bigr)\sum _{m=j}^{\infty}m^{-\alpha(q-p)-1} \\ \leq&CE|X|^{p}+C\sum_{j=1}^{\infty}j^{\alpha p}P\bigl((j-1)^{\alpha}< |X|\leq j^{\alpha}\bigr) \\ \leq&CE|X|^{p}< \infty, \end{aligned}$$
(2.13)
which yields \(I_{3}^{\prime}<\infty\).
By the fact that \(EX^{2}<\infty\) and \(E|X|^{p}<\infty\) we have for \(p\geq2\):
$$\begin{aligned} I_{4}^{\prime} \leq&C\sum_{n=1}^{\infty}n^{\alpha p-2-\alpha} \int_{n^{\alpha}}^{\infty}u^{-q}\Biggl(\sum _{i=1}^{n}EX_{i}^{2} \Biggr)^{\frac{q}{2}}\,du \\ \leq&C\sum_{n=1}^{\infty}n^{\alpha p-2-\alpha} \int_{n^{\alpha}}^{\infty}u^{-q}\bigl(n EX^{2}\bigr)^{\frac{q}{2}}\,du \\ \leq&C\sum_{n=1}^{\infty}n^{\alpha p-2-\alpha+\frac{q}{2}} \int_{n^{\alpha}}^{\infty}u^{-q}\,du \\ \leq&C\sum_{n=1}^{\infty}n^{\alpha p-2-\alpha q+\frac{q}{2}}< \infty. \end{aligned}$$
Thus, the proof of Lemma 2.4 is complete. □